Factors and Multiples

Active learning helps students visualize the relationships between factors and multiples through movement and collaboration. When students pair up or work in teams, they can test ideas together and correct misunderstandings in real time. This hands-on approach builds confidence and deepens their understanding of number patterns.

• Primary 4
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Calculate all factors for any given 2-digit number by using systematic listing.
2. Identify the lowest common multiple (LCM) of two numbers up to 50.
3. Classify numbers up to 100 as prime or composite, justifying the classification based on the number of factors.
4. Compare and contrast the definitions of factors and multiples, providing examples for each.
5. Solve word problems involving common factors and multiples.

Collaborative Problem-Solving

⏱ 25 min·Pairs

Pairs: Factor Pairs Race

Pairs receive a two-digit number and race to list all factor pairs using mini-whiteboards, checking by multiplying back. Teacher calls time after 3 minutes; pairs swap numbers and repeat twice. Discuss systematic order from 1 up.

Prepare & details

What is the difference between a factor and a multiple of a number?

Facilitation Tip: During Factor Pairs Race, remind pairs to start with 1 and its pair first to ensure systematic listing.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Small Groups: Multiples Chain Game

In groups of four, students sit in a circle and say the next multiple of two given numbers in turn, like 4 and 6. If stuck, pass a counter; first to LCM wins a point. Rotate roles and numbers for three rounds.

Prepare & details

How do you find all the factors of a 2-digit number using systematic listing?

Facilitation Tip: For the Multiples Chain Game, set a timer for one minute per round to keep the energy high and focused.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 30 min·Whole Class

Whole Class: Prime Sort Relay

Divide class into two teams. Call a number; first student from each team runs to board, states if prime or composite with factors. Correct team scores; continue for 10 numbers, then review rules.

Prepare & details

Can you identify the common multiples of two numbers and find the lowest common multiple?

Facilitation Tip: In the Prime Sort Relay, place a small timer at each station to encourage quick and accurate sorting.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 20 min·Individual

Individual: LCM Puzzle Cards

Students match pairs of numbers to their LCM cards, using factor lists as clues. Self-check with answer key, then pair to explain one match. Extend by creating own puzzles.

Prepare & details

What is the difference between a factor and a multiple of a number?

Facilitation Tip: With LCM Puzzle Cards, encourage students to use grid paper to sketch arrays if they struggle to visualize multiples.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Teach this topic by connecting factors to division and multiples to multiplication through visual models. Avoid rushing to abstract rules; instead, let students discover patterns through hands-on activities. Research shows that using physical manipulatives or quick sketches helps students internalize these concepts more deeply than rote memorization.

What to Expect

Students will confidently list factors and multiples, distinguish between prime and composite numbers, and explain how these concepts connect multiplication and division. Small group discussions and relay races will show their ability to apply these skills in problem-solving contexts.

Watch Out for These Misconceptions

Common Misconception

During Prime Sort Relay, watch for students who incorrectly label 1 as a prime number.

What to Teach Instead

Have students test 1 by trying to pair it with another factor beyond itself. Ask them to recall that prime numbers must have exactly two distinct factors, and 1 only has one. Use the sorting cards to physically move 1 out of the prime pile during the relay.

Common Misconception

During Factor Pairs Race, watch for students who confuse factors and multiples.

What to Teach Instead

Ask pairs to build rectangular arrays with grid paper for their numbers. For example, for 12, they should create arrays of 1x12, 2x6, and 3x4. This visually separates factors (the dimensions) from multiples (the total count of squares).

Common Misconception

During Multiples Chain Game, watch for students who assume the LCM is always the product of two numbers.

What to Teach Instead

Have groups use Venn diagrams on their whiteboards to list multiples side by side. Point out overlapping numbers, then ask them to divide the product by the GCF to find the LCM. This reinforces why the product isn't always the smallest shared multiple.

Assessment Ideas

Exit Ticket

After Factor Pairs Race, give each student a card with two numbers, e.g., 16 and 20. Ask them to list all factors of 16, then all factors of 20. Finally, have them identify the common factors and the LCM of the two numbers.

Quick Check

During Prime Sort Relay, write a list of numbers on the board (e.g., 7, 10, 13, 25). Ask students to hold up one finger for prime numbers and two fingers for composite numbers. Then, call on two volunteers to explain why they categorized two specific numbers.

Discussion Prompt

After Multiples Chain Game, pose this scenario: 'Liam has 42 marbles and wants to divide them equally into bags. He also has 56 buttons and wants to sort them into equal groups. What is the largest number of bags he can use so each bag has the same number of marbles? What is the largest number of buttons he can put in each group so all groups are equal?' Guide students to identify the need for common factors.

Extensions & Scaffolding

• Challenge students who finish early to find the LCM of three numbers, like 6, 8, and 12, using their puzzle cards method.
• For students who struggle, provide number lines to physically jump by a given multiple, reinforcing the skip-counting process.
• Give extra time for students to create their own factor and multiple word problems, then swap with peers to solve.

Key Vocabulary

Factor	A factor is a number that divides exactly into another number without leaving a remainder. For example, 3 is a factor of 12.
Multiple	A multiple is a number that can be divided exactly by another number. Multiples are found by skip-counting or repeated addition. For example, 24 is a multiple of 6.
Common Factor	A common factor is a number that is a factor of two or more different numbers. For example, 4 is a common factor of 12 and 20.
Lowest Common Multiple (LCM)	The lowest common multiple is the smallest positive number that is a multiple of two or more given numbers. For example, the LCM of 4 and 6 is 12.
Prime Number	A prime number is a whole number greater than 1 that has only two factors: 1 and itself. For example, 7 is a prime number.
Composite Number	A composite number is a whole number greater than 1 that has more than two factors. For example, 9 is a composite number because its factors are 1, 3, and 9.